As has been pointed out by others, the "logical" argument for *I didn't **use** to do that* is that ***did*** already carries the verb tense, so it's not needed in ***used***. Same as, for example, *I didn't **have** to do that.*

But as [this NGram][1] shows, usage has changed considerably over the past century and more...

[![enter image description here][2]][2]


Unsurprisingly therefore, you'll find plenty of traditionalist grammarians defending the older style (which is **completely inaudible** in normal speech anyway). But it's pretty obvious which version has the upper hand today, so I suggest you go with that unless you want to look like an old fuddy-duddy.

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As pointed out by John Lawler in [an ELU answer on this topic...][3]

> *Both [spellings] look bad, the first because **used** looks like a misspelled infinitive,and the second because **use to** doesn't look like it sounds like **used to** should.  
**This should be considered a bug in the orthographic system.***

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It's worth noting parallels between the voiced and unvoiced versions of ***used to / have to***. In both cases the unvoiced version has a totally different meaning (relating to *habitual action / obligation*, rather than *employment / ownership*). If you'd never seen *written* English, you'd naturally classify the voiced/unvoiced versions as ***different words***, not different pronunciations of "the same" words.

Interestingly, the "special" version of ***have to*** actually does have two different pronunciations itself (past tense *I **had** to go then* is pronounced ***hat***; in present tense *I **have** to go now* it's ***haff***). There's no such split with ***used to*** because in semantic terms it's always "past tense" anyway. And since language use is primarily driven by spoken rather than written forms, this is essentially why people increasingly tend to ignore the pedagogic / logical arguments and stick with ***used to*** in all contexts.

  [1]: https://books.google.com/ngrams/graph?content=didn%27t%20use%20to%2Cdidn%27t%20used%20to&year_start=1900&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cdid%20not%20use%20to%3B%2Cc0%3B.t1%3B%2Cdid%20not%20used%20to%3B%2Cc0
  [2]: https://i.sstatic.net/rNstk.png
  [3]: https://english.stackexchange.com/a/314973/2637